In this chapter the goal of obtaining the “local Lyapunov exponents” as sublimiting exponential growth rates is tackled. As already described, the system under consideration is the real-noise driven stochastic system
where A ∈ C(ℝd,Kn×n) is a continuous matrix function (K = ℝ or ℂ), d ∈ ℕ and n ∈ ℕ are the dimensions of the state spaces of Xε and Zε, respectively, ε ≥ 0 parametrizes the intensity of (W t )t≥0 which denotes a Brownian motion in ℝd on a complete probability space (Ω,F, ℙ) and Xε,ξ is a diffusion starting in ξ ∈ ℝd, defined by the SDE (2.1) such that the assumptions 2.1.1 hold. For Zε, solving the random vector differential equation
we will use the equivalent notations
as before, where
solves the random matrix differential equation
The object of interest is the exponential growth rate
on the time scale T(ε). Any limit as ε → 0 of this rate will be called local Lyapunov exponent of Zε.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2009). Local Lyapunov exponents. In: Local Lyapunov Exponents. Lecture Notes in Mathematics, vol 1963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85964-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-85964-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85963-5
Online ISBN: 978-3-540-85964-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)